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Book DetailsMathematical Surveys and MonographsVolume: 217; 2017; 704 ppMSC: Primary 55; Secondary 18; 57; 20
The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.
The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2disc operad.
This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
ReadershipGraduate students and researchers interested in algebraic topology and algebraic geometry.
This item is also available as part of a set: 
Table of Contents

Homotopy theory and its applications to operads

General methods of homotopy theory

Model categories and homotopy theory

Mapping spaces and simplicial model categories

Simplicial structures and mapping spaces in general model categories

Cofibrantly generated model categories

Modules, algebras, and the rational homotopy of spaces

Differential graded modules, simplicial modules, and cosimplicial modules

Differential graded algebras, simplicial algebras, and cosimplicial algebras

Models for the rational homotopy of spaces

The (rational) homotopy of operads

The model category of operads in simplicial sets

The homotopy theory of (Hopf) cooperads

Models for the rational homotopy of (nonunitary) operads

The homotopy theory of (Hopf) $\Lambda $cooperads

Models for the rational homotopy of unitary operads

Applications of the rational homotopy to $E_n$operads

Complete Lie algebras and rational models of classifying spaces

Formality and rational models of $E_n$operads

The computation of homotopy automorphism spaces of operads

Introduction to the results of the computations for the $E_n$operads

The applications of homotopy spectral sequences

Homotopy spsectral sequences and mapping spaces of operads

Applications of the cotriple cohomology of operads

Applications of the Koszul duality of operads

The case of $E_n$operads

The applications of the Koszul duality for $E_n$operads

The interpretation of the result of the spectral sequence in the case of $E_2$operads

Conclusion: A survey of further research on operadic mapping spaces and their applications

Graph complexes and $E_n$operads

From $E_n$operads to embedding spaces

Appendices

Cofree cooperads and the bar duality of operads


Additional Material

Reviews

This book provides a very useful reference for known and new results about operads and rational homotopy theory and thus provides a valuable resource for researchers and graduate students interested in (some of) the many topics that it covers. As it is the case for the first volume, careful introductions on the various levels of the text help to make this material accessible and to put it in context.
Steffen Sagave, Zentralblatt MATH


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The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads.
The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2disc operad.
This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
Graduate students and researchers interested in algebraic topology and algebraic geometry.

Homotopy theory and its applications to operads

General methods of homotopy theory

Model categories and homotopy theory

Mapping spaces and simplicial model categories

Simplicial structures and mapping spaces in general model categories

Cofibrantly generated model categories

Modules, algebras, and the rational homotopy of spaces

Differential graded modules, simplicial modules, and cosimplicial modules

Differential graded algebras, simplicial algebras, and cosimplicial algebras

Models for the rational homotopy of spaces

The (rational) homotopy of operads

The model category of operads in simplicial sets

The homotopy theory of (Hopf) cooperads

Models for the rational homotopy of (nonunitary) operads

The homotopy theory of (Hopf) $\Lambda $cooperads

Models for the rational homotopy of unitary operads

Applications of the rational homotopy to $E_n$operads

Complete Lie algebras and rational models of classifying spaces

Formality and rational models of $E_n$operads

The computation of homotopy automorphism spaces of operads

Introduction to the results of the computations for the $E_n$operads

The applications of homotopy spectral sequences

Homotopy spsectral sequences and mapping spaces of operads

Applications of the cotriple cohomology of operads

Applications of the Koszul duality of operads

The case of $E_n$operads

The applications of the Koszul duality for $E_n$operads

The interpretation of the result of the spectral sequence in the case of $E_2$operads

Conclusion: A survey of further research on operadic mapping spaces and their applications

Graph complexes and $E_n$operads

From $E_n$operads to embedding spaces

Appendices

Cofree cooperads and the bar duality of operads

This book provides a very useful reference for known and new results about operads and rational homotopy theory and thus provides a valuable resource for researchers and graduate students interested in (some of) the many topics that it covers. As it is the case for the first volume, careful introductions on the various levels of the text help to make this material accessible and to put it in context.
Steffen Sagave, Zentralblatt MATH